Page - 209 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Comments. (i) Wetake intoaccounttheglobalprobabilitydensity p.Thenanobjectof thecategoryGM(Ξ,Ω) isdenotedby [E,π,M,D,p]. (ii) The function p isΓ-equivariant. THISISTHEGEOMETRYinthe senseofErlangenprogram. (iii)Wehavenotusedanyargumentdepending thedimensionofmanifolds. TheFigure5expresses coherence to localprobabilitydensitiesWeare inposition todefine themorphismsof the categoryGM(Ξ,Ω). B1A1 C1 R A B C φi π Φi γij γij φj Φj Pi Pj p1 p1 Figure5.Localisation. Ei E R Pi p Figure6.ProbabilityDensity. InFigure5oneseesthatmodulothedynamicsof thegroupΓ inRm×Ξall localizationslookalike. Figure6showthat localprobabilitydensities{pi}arebut localizationsofaglobalprobabilitydensityp 8.3.3. TheMorphismsofGM(Ξ,Ω) Definition 51. LetM = [E,π,M,D,p] andM∗ = [E∗,π∗,M∗,D∗,p∗] be two objects of the category GM(Ξ,Ω). AFB(Γ,Ξ)-morphism (Ψ×ψ) : [E,π,M,D]→ [E∗,π∗,M∗,D∗] 209